Intuitionistic

Intuitionistic refers to a tradition in logic and mathematics that arose from the philosophical views of L. E. J. Brouwer and was formalized by Arend Heyting. It contrasts with classical mathematics by rejecting the law of the excluded middle as a general principle. In intuitionistic logic, a statement is considered true only if there is a constructive proof, and an existence claim must provide a method to construct an example. Consequently, a disjunction A or B is true only when one has a proof of A or a proof of B, and not merely a nonconstructive argument.

The formal foundation of intuitionistic logic is the intuitionistic propositional calculus developed by Heyting, along with intuitionistic arithmetic. The semantics are often given by Kripke models or Beth models, where truth is persistent and can grow as more information becomes available. In these frameworks, proving the negation of a statement means showing that assuming the statement leads to a contradiction with a constructive proof.

Beyond syntax, intuitionistic logic appears as the internal logic of any topos, giving a categorical interpretation in which truth values form a Heyting algebra rather than a Boolean algebra. The Curry–Howard correspondence links intuitionistic proofs with programs, aligning logical deduction with typed lambda calculi and computational content.

Intuitionistic mathematics has influenced constructive analysis and computer science, notably in proof assistants and programming languages based on constructive type theories. Systems such as Coq, Agda, and Lean embody these ideas, enabling the extraction of executable programs from proofs. While closely related to constructive mathematics, intuitionistic logic emphasizes the necessity of explicit constructions in proving mathematical statements.